1. LDK R Logics for Data and Knowledge Representation Modal Logic Originally by Alessandro Agostini and Fausto Giunchiglia Modified by Fausto Giunchiglia, Rui Zhang and Vincenzo Maltese

2. 2 Outline  Introduction  Syntax  Semantics  Satisfiability and Validity  Kinds of frames  Correspondence with FOL 2

3. Introduction  We want to model situations like this one: 1. “Fausto is always happy” circumstances” 2. “Fausto is happy under certain  In PL/ClassL we could have: HappyFausto  In modal logic we have: 1. □ HappyFausto 2. ◊ HappyFausto As we will see, this is captured through the notion of “possible words” and of “accessibility relation” 3

4. Syntax  We extend PL with two logical modal operators: □ (box) and ◊ (diamond) □ P : “Box P” or “necessarily P” or “P is necessary true” ◊P : “Diamond P” or “possibly P” or “P is possible” Note that we define □ P =  ◊  P, i.e. □ is a primitive symbol  The grammar is extended as follows: ::= A | B |... | P | Q |... | ⊥ | ⊤ | ::= | ¬ | ∧ | ∨ |  |  | □ | ◊ 4

5. Different interpretations 5 Philosophy □ P : “P is necessary” ◊P : “P is possible” Epistemic □ a P : “Agent a believes P ” or “Agent a knows P” Temporal logics □ P : “P is always true” ◊P : “P is sometimes true” Dynamic logics or logics of programs □ a P : “P holds after the program a is executed” Description logics □ HASCHILD MALE  ∀ HASCHILD.MALE ◊ HASCHILD MALE  ∃ HASCHILD.MALE

6. Semantics: Kripke Model  A Kripke Model is a triple M = where:  W is a non empty set of worlds  R ⊆ W x W is a binary relation called the accessibility relation  I is an interpretation function I: L  pow(W) such that to each proposition P we associate a set of possible worlds I(P) in which P holds  Each w ∈ W is said to be a world, point, state, event, situation, class … according to the problem we model  For "world" we mean a PL model. Focusing on this definition, we can see a Kripke Model as a set of different PL models related by an "evolutionary" relation R; in such a way we are able to represent formally - for example - the evolution of a model in time.  In a Kripke model, is called frame and is a relational structure. 6

7. Semantics: Kripke Model  Consider the following situation:  M = W = {1, 2, 3, 4} R = {,,,, } I(BeingHappy) = {2} I(BeingSad) = {1} I(BeingNormal) = {3, 4} 7 123 4 BeingHappy BeingSad BeingNormal

8. Truth relation (true in a world)  Given a Kripke Model M =, a proposition P ∈ L ML and a possible world w ∈ W, we say that “w satisfies P in M” or that “P is satisfied by w in M” or “P is true in M via w”, in symbols: M, w ⊨ P in the following cases: 1. P atomicw ∈ I(P) 2. P =  QM, w ⊭ Q 3. P = Q  TM, w ⊨ Q and M, w ⊨ T 4. P = Q  TM, w ⊨ Q or M, w ⊨ T 5. P = Q  TM, w ⊭ Q or M, w ⊨ T 6. P = □ Q for every w’ ∈ W such that wRw’ then M, w’ ⊨ Q 7. P = ◊ Q for some w’ ∈ W such that wRw’ then M, w’ ⊨ Q NOTE: wRw’ can be read as “w’ is accessible from w via R” 8

9. Semantics: Kripke Model  Consider the following situation:  M = W = {1, 2, 3, 4} R = {,,,, } I(BeingHappy) = {2} I(BeingSad) = {1} I(BeingNeutral) = {3, 4} M, 2 ⊨ BeingHappy M, 2 ⊨  BeingSad M, 4 ⊨ □ BeingHappy M, 1 ⊨ ◊BeingHappy M, 1 ⊨  ◊BeingSad 9 123 4 BeingHappy BeingSad BeingNormal

10. Satisfiability and Validity  Satisfiability A proposition P ∈ L ML is satisfiable in a Kripke model M = if M, w ⊨ P for all worlds w ∈ W. We can then write M ⊨ P  Validity A proposition P ∈ L ML is valid if P is satisfiable for all models M (and by varying the frame ). We can write ⊨ P 10

11. Satisfiability  Consider the following situation:  M = W = {1, 2, 3, 4} R = {,,, } I(BeingHappy) = {2} I(BeingSad) = {1} I(BeingNormal) = {3, 4} M, w ⊨ □ BeingHappy for all w ∈ W, therefore □ BeingHappy is satisfiable in M. 11 123 4 BeingHappy BeingSad BeingNormal

12. Validity  Prove that P: □ A  ◊A is valid  In all models M =, (1) □ A means that for every w ∈ W such that wRw’ then M, w’ ⊨ A (2) ◊A means that for some w ∈ W such that wRw’ then M, w’ ⊨ A It is clear that if (1) then (2) in the example (as we will see this is valid in serial frames) 12 12 3 A A

13. Kinds of frames  Given the frame F =, the relation R is said to be:  Serial iff for every w ∈ W, there exists w’ ∈ W s.t. wRw’  Reflexive iff for every w ∈ W, wRw  Symmetric iff for every w, w’ ∈ W, if wRw’ then w’Rw  Transitiveiff for every w, w’, w’’ ∈ W, if wRw’ and w’Rw’’ then wRw’’  Euclidianiff for every w, w’, w’’ ∈ W, if wRw’ and wRw’’ then w’Rw’’  We call a frame serial, reflexive, symmetric or transitive according to the properties of the relation R 13

14. Kinds of frames  Serial: for every w ∈ W, there exists w’ ∈ W s.t. wRw’  Reflexive: for every w ∈ W, wRw  Symmetric: for every w, w’ ∈ W, if wRw’ then w’Rw 14 123 12 12 3

15. Kinds of frames  Transitive: for every w, w’, w’’ ∈ W, if wRw’ and w’Rw’’ then wRw’’  Euclidian: for every w, w’, w’’ ∈ W, if wRw’ and wRw’’ then w’Rw’’ 15 123 12 3

16. Valid schemas  A schema is a formula where I can change the variables  THEOREM. The following schemas are valid in the class of indicated frames: D: □ A  ◊ Avalid for serial frames T: □ A  Avalid for reflexive frames B:A  □ ◊ Avalid for symmetric frames 4: □ A  □□ Avalid for transitive frames 5: ◊ A  □ ◊ Avalid for Euclidian frames NOTE: if we apply T, B and 4 we have an equivalence relation  THEOREM. The following schema is valid: K: □ (A  B)  ( □ A  □ B)Distributivity of □ w.r.t.  16

17. Proof for T : □ A  A valid for reflexive frames Assuming M, w ⊨ □ A, we want to prove that M, w ⊨ A. From the assumption M, w ⊨ □ A, we have that for every w’ ∈ W such that wRw’ we have that M, w’ ⊨ A (1). Since R is reflexive we also have w’Rw, we then imply that M, w ⊨ A (by substituting w to w’ in (1)) 17 □A, A 12

18. Proof for B : A  □◊A valid for symmetric frames Assume M, w ⊨ A. To prove that M, w ⊨ □ ◊ A we need to show that for every w’ ∈ W such that wRw’ then M, w ⊨ ◊ A. M, w ⊨ ◊ A is that for some w’’ ∈ W such that w’Rw’’ then M, w’’ ⊨ A. Therefore we need to prove that for every w’ ∈ W such that wRw’ and for some w’’ ∈ W such that w’Rw’’ then M, w’’ ⊨ A Since R is symmetric, from wRw’ it follows that w’Rw. For w’’ ∈ W such that w’’ = w, we have that w’Rw’’ and M, w’’ ⊨ A. Hence M, w ⊨ A. 18 12 3 A, □◊A ◊A

19. Reasoning services: EVAL  Model Checking (EVAL) Given a (finite) model M = and a proposition P ∈ L ML we want to check whether M, w ⊨ P for all w ∈ W M, w ⊨ P for all w ? 19 EVAL M, P Yes No

20. Reasoning services: SAT  Satisfiability (SAT) Given a proposition P ∈ L ML we want to check whether there exists a (finite) model M = such that M, w ⊨ P for all w ∈ W Find M such that M, w ⊨ P for all w 20 SAT P M No

21. Reasoning services: UNSAT  Unsatisfiability (unSAT) Given a (finite) model M = and a proposition P ∈ L ML we want to check that does not exist any world w such that M, w ⊨ P Verify that  ∃ w such that M, w ⊨ P 21 VAL M,  P w No

22. Reasoning services: VAL  Validity (VAL) Given a a proposition P ∈ L ML we want to check that M, w ⊨ P for all (finite) models M = and w ∈ W Verify that M, w ⊨ P for all M and w 22 VAL P Yes No

23. Correspondence between □ and ∀ ( ◊ and ∃ )  We can define a translation function T: L ML  L FO as follows: 1. T(P) = P(x) for all propositions P in L ML 2. T(  P) =  T(P) for all propositions P 3. T(P * Q) = T(P) * T(Q) for all propositions P, Q and * ∈ { , ,  } 4. T( □ P) = ∀ x T(P) for all propositions P 5. T( ◊ P) = ∃ x T(P) for all propositions P THEOREM: For all propositions P in L ML, P is modally valid iff T(P) is valid in FOL. 23